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To take an example from a field in which Poincare did some
of his most original work, consider a homogeneous, incompres-
sible fluid mass held together by the gravitation of its particles
and rotating about an axis. Under what conditions will the
motion be stable and what will be the possible shapes of such a
stably rotating fluid? MacLaurin, Jacobi, and others proved
that certain ellipsoids will be stable; Poincare, using more
intuitive, 'less arithmetical' methods than his predecessors,
once thought he had determined the criteria for the stability of
a pear-shaped body. But he had made a slip. His methods were
not adapted to numerical computation and later workers,
including G. H. Darwin, son of the famous Charles, undeterred
by the horrific jungles of algebra and arithmetic that must be
cleared out of the way before a definite conclusion can be
reached, undertook a decisive solution.*
The man interested in the evolution of binary stars is more
comfortable if the findings of the mathematicians are presented
to him in a form to which he can apply a calculating machine.
And since Kronecker's fiat of 'no construction, no existence',
some pure mathematicians themselves have been less enthu-
siastic than they were in Poineare's day for existence theorems
which are not constructive. Poincare^s scorn for the kind of
detail that users of mathematics demand and must have before
they can get on with their work was one of the most important
contributory causes to his universality. Another was his extra-
ordinarily comprehensive grasp of all the machinery of the
theory of functions of a complex variable. In this he had no
equal. And it may be noted that Poincare turned his universa-
lity to magnificent use in disclosing hitherto unsuspected con-
nexions between distant branches of mathematics, for example
between (continuous) groups and linear algebra.
* This famous question of the 'piriform body', of considerable
importance in cosmogony, was apparently settled in 190S by Liapou-
noff, whose conclusion was confirmed in 1915 by Sir James Jeans:
they found that the motion is unstable. Few have had the courage to
check the calculations. After 1915 Leon Lichtenstein, a fellow-
countryman of Liapounoff, made a general attack on the problem of
rotating fluid masses. The problem seems to be unlucky; both L*s
had violent deaths.
M.M.—VOL. u                                 K                                   583